Andrew Nealen, TU Berlin, CG 11 Andrew Nealen TU Berlin Takeo Igarashi The University of Tokyo / PRESTO JST Olga Sorkine Marc Alexa TU Berlin Laplacian Mesh Optimization

Andrew Nealen, TU Berlin, CG 22 What is it ?

Andrew Nealen, TU Berlin, CG 33 Overview  Motivation Problem formulation Laplacian mesh processing basics  Laplacian mesh optimization framework  Applications Triangle shape optimization Mesh smoothing  Discussion

Andrew Nealen, TU Berlin, CG 44 Motivation  Local detail preserving triangle optimization A Sketch-Based Interface for Detail Preserving Mesh Editing [Nealen et al. 2005]

Andrew Nealen, TU Berlin, CG 55 Motivation  Local detail preserving triangle optimization A Sketch-Based Interface for Detail Preserving Mesh Editing [Nealen et al. 2005]  Can we perform global optimization this way ? = L x 

Andrew Nealen, TU Berlin, CG 66 Laplacian Mesh Processing  Discrete Laplacians = Lx  n  cotangent : w ij = cot  ij + cot  ij  uniform : w ij = 1

Andrew Nealen, TU Berlin, CG 77 Laplacian Mesh Processing  Surface reconstruction n  cotangent : w ij = cot  ij + cot  ij  uniform : w ij = 1 = Lx  LL y z x zz yy xx

Andrew Nealen, TU Berlin, CG 88 Laplacian Mesh Processing  Surface reconstruction n zz yy xx y z x = L L Lc1c1    fix edit c2c2   

Andrew Nealen, TU Berlin, CG 99 Laplacian Mesh Processing  Least-squares solution n zz yy xx y z x = L L Lc1c1    fix edit c2c2    w1w1 w1w1 w2w2 w2w2 w Li Ax = b ATAATAx = bATAT (A T A) -1 x = bATAT Normal Equations

Andrew Nealen, TU Berlin, CG 10 Laplacian Mesh Processing  Tangential smoothing n zz yy xx y z x = L L L fix c1c1    L L L

Andrew Nealen, TU Berlin, CG 11 L L L Laplacian Mesh Processing  Tangential smoothing n zz yy xx y z x = fix c1c1   

Andrew Nealen, TU Berlin, CG 12 L L L Laplacian Mesh Processing  Tangential smoothing n zz yy xx y z x = fix c1c1   

Andrew Nealen, TU Berlin, CG 13 More motivation…  So: can we use such a system for global optimization ? =Lx 

Andrew Nealen, TU Berlin, CG 14 Our Solution  All vertices are (weighted) anchors  Preserves global shape  Uses existing LS framework  Anchor + Laplacian weights determine result

Andrew Nealen, TU Berlin, CG 15 Framework  Detail preserving tri shape optimization for L = L uni and f =  cot  (similar to local optimization)  Mesh smoothing L = L cot (outer fairness) or L = L uni (outer and inner fairness) and f = 0 = Lxf WLWL WLWL p WPWP WPWP

Andrew Nealen, TU Berlin, CG 16 Tri Shape Optimization  Detail preserving tri shape optimization = L uni x  p WPWP WPWP

Andrew Nealen, TU Berlin, CG 17 Positional Weights

Andrew Nealen, TU Berlin, CG 18 Constant Weights

Andrew Nealen, TU Berlin, CG 19 Linear Weights

Andrew Nealen, TU Berlin, CG 20 CDF Weights

Andrew Nealen, TU Berlin, CG 21 CDF Weights

Andrew Nealen, TU Berlin, CG 22 Sharp Features

Andrew Nealen, TU Berlin, CG 23 Sharp Features

Andrew Nealen, TU Berlin, CG 24 Sharp Features

Andrew Nealen, TU Berlin, CG 25 Mesh Smoothing  Mesh smoothing L = L cot (outer fairness) or L = L umb (outer and inner fairness) and f = 0  Controlled by W P and W L (Intensity, Features)  Similar to Least-Squares Meshes [Sorkine et al. 04] = Lx0 WLWL WLWL p WPWP WPWP

Andrew Nealen, TU Berlin, CG 26 Using W P

Andrew Nealen, TU Berlin, CG 27 Using W P and W L

Andrew Nealen, TU Berlin, CG 28 Results

Andrew Nealen, TU Berlin, CG 29 Noisy

Andrew Nealen, TU Berlin, CG 30 Smoothed

Andrew Nealen, TU Berlin, CG 31 Original

Andrew Nealen, TU Berlin, CG 32 Tri Shape Optimization

Andrew Nealen, TU Berlin, CG 33 Smoothing Outer and Inner Fairness (L umb )

Andrew Nealen, TU Berlin, CG 34 Original

Andrew Nealen, TU Berlin, CG 35 Tri Shape Optimization

Andrew Nealen, TU Berlin, CG 36 Smoothing Outer Fairness only (L cot )

Andrew Nealen, TU Berlin, CG 37 Discussion  The good... Easily controllable tri shape optimization and smoothing Leverages existing least squares framework Can replace tangential smoothing step for general remeshers ... and the not so good Euclidean distance is not Hausdorff distance, so error control is indirect Does rely on some (user) parameter tweaking

Andrew Nealen, TU Berlin, CG 38 Thank you !  Contact info Andrew Nealen Takeo Igarashi Olga Sorkine Marc Alexa